Fly-half Stephen Jones and young centre Jonathan Davies played key roles as Nigel Davies' men, who face the Ospreys next week, did most of the damage in the opening 40 minutes.
Teenage winger George North, Davies and new captain Matthew Rees helped themselves to tries, leaving the Dragons trailing by 24-3 at half-time.
Davies, Rees and Tavis Knoyle were all sin-binned, either side of the break, and the Dragons got the scoreboard ticking thanks to two Matthew Jones penalties and a try from full-back Will Harries.
However, the Scarlets kept the hosts scoreless thereafter and Jones booted his second penalty goal to rubberstamp a deserved away victory.
There was huge commitment from both sides, some hard hits throughout and no lack of skill as these Welsh rivals looked to lay down an early season marker.
The Dragons had their regular captain Tom Willis back at hooker, and their other team changes from last weekend's derby defeat to Cardiff saw winger Adam Hughes and prop Phil Price called up.
The Scarlets management made two changes to the side that started the six-try dismissal of Aironi Rugby. The wily Vernon Cooper was restored to the second row, with Josh Turnbull brought in at blindside flanker.
Winless in their last four encounters with the Scarlets, the men of Gwent hit the front in the seventh minutes when Matthew Jones split the posts with a penalty.
But the wind-backed Scarlets' response was immediate. Set up by Davies, North took advantage of some slack defending to nab his fourth try in four league games. Stephen Jones stepped up to add the conversion.
The game swung further in the Scarlets' favour when Dragons flanker Robin Sowden-Taylor was sin-binned for not rolling away after the tackle, allowing Jones to kick his side into a 10-3 lead.
They tagged on two more converted tries before the break, from centre Davies and hooker Rees, to give the Dragons a mountain to climb at 24-3 down.
Davies' try was a cracker as he broke past Matthew Jones galloped over from the ten-metre line, reaching his destination despite a last ditch tackle from Hughes.
Taking a pass from Deacon Manu, Rees managed to wriggle his way over from close range, which television match official Neil Ballard confirmed to referee James Jones.
Still, quick-fire yellow cards for both try scorers gave the hosts some hope.
Predictably, Paul Turner's side got on top in the third quarter. Number 8 Gavin Thomas unfortunately fumbled the ball as he looked set to touch down, before Matthew Jones kicked a brace of penalties.
The game was turning into a real dog fight. A try from Harries, who scored in the left corner off a Tom Riley pass, closed the gap to 24-14. It was the full-back's third of the campaign.
Just prior to that, Scarlets scrum half Knoyle had been flashed a yellow card for tackling his opposite number Wayne Evans off the ball.
But Stephen Jones, who put in another assured performance, found the target with a final penalty to confirm the Scarlets' fifth ever league win at Rodney Parade, one more than the Blues.
The in-form Davies had to hobble off with an ankle injury in the closing stages, but admitted afterwards that it was just a sprain.
Flanker Danny Lydiate was the standout player for the Dragons whose season so far has been hampered by injuries. They face a tricky trip to Benetton Treviso in the next round.
|Newport Gwent Dragons Score Card|
|Scarlets Score Card|